Mechanical Vibrations

Protecting Against a Shaky Floor — Vibration Isolation

When the base shakes, the response ratio is the transmissibility TR=X/Y; isolation (TR<1) needs r>√2, so lower ωₙ (soft mount) — and damping is double-edged

A4's resonance pushed the mass directly, but in the real world the floor shakes — road bumps, earthquakes, a machine next door. What matters is the transmissibility TR = XY, how much the mass shakes (X) when the base shakes (Y) — and, surprisingly, the shaking only starts to shrink (isolation) once the driving frequency exceeds √2 times the natural frequency.

Instead of forcing the mass, shake the floor. Change the drive ratio r = ωωₙ and three regimes appear: slow (r≪1) and the mass just follows the base (TR≈1); near r=1 resonance makes it shake even more (TR≫1); fast (large r) and the mass stays almost still (TR<1, isolation).

The transmissibility curve is the map. Drag r. A resonance peak rises at r=1, and at r=√2 the curve passes exactly through TR=1 (whatever the damping). Only to its right, the green region r>√2, is TR<1 — the true isolation zone.

So how do you isolate? The disturbance frequency is fixed, so you make the mount soft (lower stiffness k → lower ωₙ) to push r past √2. Lower the stiffness — you pass through resonance (r=1) with a big swing on the way, but go softer still and the mass calms down. The secret to isolation is a soft mount.

Damping is a double-edged sword. Raise the damping ratio ζ: the resonance peak drops, but in the isolation region (r>√2) the TR actually rises — isolation gets worse. The proof is that all the curves meet at the single point r=√2, TR=1. So a mount is given just enough damping to survive resonance at startup and shutdown, and no more.

The same transmissibility rules both directions. Flip through the examples — a car suspension (road → body), an engine or washing-machine mount (machine → floor, now a force transmissibility), an isolation table for instruments and microscopes (floor → equipment). Whether you protect the mass from the floor or the floor from the machine, it is the same curve and the same prescription — lower ωₙ.

In PracticeVibration isolation is the cousin of resonance (A4), except the base shakes, not a force. The key is the transmissibility TR = XY = √[(1+(2ζr)²)((1−r²)²+(2ζr)²)], with r = ωωₙ. For r≪1, TR≈1 (the mass follows); at r=1, resonance; at r=√2, TR=1 regardless of damping; and only for r>√2 is TR<1 (isolation). So the golden rule of isolation is to make ωₙ low enough — a soft mount that pushes the drive frequency above √2·ωₙ. Damping is double-edged: it lowers the resonance peak but worsens isolation, so use just enough to pass through resonance at startup and shutdown. Displacement transmissibility equals force transmissibility, so protecting equipment from a shaky floor and keeping a machine's vibration out of the floor are the same problem. Car suspensions, engine mounts, washing machines, earthquake base isolation, and microscope tables all live on one curve.

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